The total integral is the sum of $n$ individual trapezoids: $I \approx \frac{h}{2} \left[ f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]$ Global Error Bound: Because errors from individual segments accumulate, the total global error $E_a$ is bounded by the second derivative maximum over the entire domain $[a,b]$: $|E_a| \le \frac{(b-a)^3}{12n^2} \max_{x \in [a,b]} |f''(x)| = \frac{(b-a)h^2}{12} \max_{x \in [a,b]} |f''(x)|$ This demonstrates that the composite rule has an overall truncation error of $O(h^2)$.

Uses a parabola (second-order polynomial) connecting three points. The error is proportional to the fourth derivative, making it exact for polynomials up to the third degree (cubics). $I \approx \frac{h}{3} [f(x_0) + 4f(x_1) + f(x_2)]$ Multiple-Application Form: Requires an even number of segments (an odd number of points). $I \approx \frac{h}{3} \left[ f(x_0) + 4\sum_{i=1,3,5}^{n-1} f(x_i) + 2\sum_{j=2,4,6}^{n-2} f(x_j) + f(x_n) \right]$ Global Error Bound: The composite $1/3$ rule error is strictly bounded by: $|E_a| \le \frac{(b-a)^5}{180n^4} \max_{x \in [a,b]} |f^{(4)}(x)| = \frac{(b-a)h^4}{180} \max_{x \in [a,b]} |f^{(4)}(x)|$ Thus, Simpson's 1/3 rule has a high accuracy of order $O(h^4)$.

Uses a cubic curve (third-order polynomial) connecting four points. It also has a truncation error of $O(h^4)$ globally, and is primarily used when there is an odd number of segments, often combined with the 1/3 rule for the remaining even segments. $I \approx \frac{3h}{8} [f(x_0) + 3f(x_1) + 3f(x_2) + f(x_3)]$

Adaptive methods dynamically adjust the step size $h$ based on an estimated local error. The algorithm computes the integral over a specific interval using two methods (e.g., a basic Simpson's rule and a refined Simpson's rule with twice the segments). If the absolute difference exceeds a specified tolerance, the interval is recursively halved in the problematic region. This concentrates computational effort only where the integrand is "difficult," drastically improving overall efficiency.

The most common form uses the roots of Legendre polynomials. To apply the method, the limits of integration must be algebraically transformed from the original domain $[a, b]$ to the standardized interval $[-1, 1]$ using a linear change of variables: Change of Limits Formula: Let $x = \frac{b+a}{2} + \frac{b-a}{2}t$, which implies $dx = \frac{b-a}{2}dt$. The integral becomes a weighted sum evaluated at the roots $t_i$: $\int_{a}^{b} f(x) dx \approx \frac{b-a}{2} \sum_{i=1}^n c_i f\left( \frac{b+a}{2} + \frac{b-a}{2}t_i \right)$ Where $c_i$ are the optimal weights and $t_i$ are the roots of the $n$-th degree Legendre polynomial. An $n$-point formula perfectly integrates polynomials up to degree $2n-1$, making it exceptionally efficient for smooth functions. Explicit Values:

Monte Carlo integration uses random uniform sampling to approximate the integral. By randomly sampling $N$ points $\mathbf{x}_i$ within the multidimensional integration domain $V$, the integral is approximated as the average function value multiplied by the total domain volume: $\int_V f(\mathbf{x}) d\mathbf{x} \approx V \cdot \frac{1}{N} \sum_{i=1}^N f(\mathbf{x}_i)$ While the convergence rate is slow ($O(1/\sqrt{N})$), it is fundamentally independent of the number of dimensions. Therefore, for very high-dimensional problems (common in statistical mechanics, quantum physics, and quantitative finance), Monte Carlo methods remain the only computationally feasible approach.